M3220-1 Third Homework Assignment

MATH 3220 § 1 THIRD HOMEWORK ASSIGNMENT Due Thursday,

A. Treibergs February 10, 2005

MATH 3220 § 1	THIRD HOMEWORK ASSIGNMENT	Due Thursday,
A. Treibergs		February 10, 2005

Please hand in the following exercises from the text "An Introduction to Analysis, Third Edition" by William Wade, Prentice Hall, 2004.

195 [ 1b, 2, 5 ].

Please hand in the additional exercises.

Can a continuous function be the uniform limit of a sequence of discontinuous functions?

Suppose that the power series s(x) = Σ_k=0^∞ a_k x^k converges uniformly on (-x₀,x₀) where x₀>0. Show that s(x) is differentiable on (-x₀,x₀) and that s'(x) = Σ_k=1^∞ k a_k x^k-1.

Please hand in the following exercises from the text "An Introduction to Analysis, Third Edition" by William Wade, Prentice Hall, 2004.
195 [ 1b, 2, 5 ].
Please hand in the additional exercises.
Can a continuous function be the uniform limit of a sequence of discontinuous functions? Suppose that the power series s(x) = Σ_k=0^∞ a_k x^k converges uniformly on (-x₀,x₀) where x₀>0. Show that s(x) is differentiable on (-x₀,x₀) and that s'(x) = Σ_k=1^∞ k a_k x^k-1.